Binary neutron-star mergers: a review of Einstein's richest laboratory

The merger of binary neutron-stars systems combines in a single process: extreme gravity, copious emission of gravitational waves, complex microphysics, and electromagnetic processes that can lead to astrophysical signatures observable at the largest redshifts. We review here the recent progress in understanding what could be considered Einstein's richest laboratory, highlighting in particular the numerous significant advances of the last decade. Although special attention is paid to the status of models, techniques, and results for fully general-relativistic dynamical simulations, a review is also offered on initial data and advanced simulations with approximate treatments of gravity. Finally, we review the considerable amount of work carried out on the post-merger phase, including: black-hole formation, torus accretion onto the merged compact object, connection with gamma-ray burst engines, ejected material, and its nucleosynthesis.